Core 4 vectors question? No solution available on physicsandmathstutor

The position vectors of the points A and B relative to the origin, O, are

5i+ 4j+ k

-i+j-2k respectively

Find the position vector of the point p which lives on AB such that AP= 2BP

Find the vector A->B.
Divide it by three, and add it to A.

Thank you xo

Do you understand why? I hope you do. Let me know if you don't.

Tbf it requires basic vector understanding and I would assume your teacher is not providing this.

Erm, Well, I did what you said, and its not right:

Vector AB= (-1,1,-2) - (5,4,1) = (-6,-3,-3)

(-6,-3,-3) divided by 3= (-2,-1,-1)

O to P= ( 5,4,1)+ (-2,-1,-1)= (3,3,0)

disclaimer: not completely sure
but if AP=2BP and P is a vector xi+yj+zk then x-A(i)=2(x-B(i)), y-A(j)=2(y-B(j))
and z-A(k)=2(y-B(k)). you can now solve for x y and z where A(i) is the component of the vector A in the i direction, A(k) is the component of the vector A in the k direction, etc (same idea for B).
Could you tell me if this gets you the right answer or not? I'm also learning

What I think:

AP = 2BP

Therefore: p - a = 2(p - b)
p - a = 2p - 2b
p = 2b - a

Therefore: p = 2(-1, 1, -2) - (5, 4, 1)
p = (-2,2,-4) - (5,4,1)
p = (-7, -2, -5)